1. Introduction
Exploring second-order weak interaction nuclear transitions represents a crucial channel in addressing unresolved challenges within modern physics. The detection of neutrinoless double beta decay (DBD) holds particular significance, as it would confirm the Majorana nature of neutrinos [
1] (meaning they are their own antiparticles), and potentially provide insights into the mechanism generating their mass, as well as into charge-parity (CP) violation in the lepton sector [
2,
3,
4,
5,
6,
7,
8]. Moreover, it would mark the first instance of the detection of a lepton number-violating process in laboratory. These types of beyond the Standard Model (SM) processes might play a significant role in certain leptogenesis mechanisms, which may elucidate the prevalence of matter over antimatter in the Universe [
9,
10].
Currently, the most studied modes of DBD involve the emission of two electrons from neutron-rich candidate isotopes: the two-neutrino mode (
) and the neutrinoless mode (
). While the latter remains experimentally elusive [
11,
12,
13,
14,
15,
16,
17,
18], the former serves as a playground to test different hypotheses and the predictive strengths of the nuclear structure models [
19,
20,
21], and to constrain various parameters associated with physics beyond the SM [
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32]. A comprehensive and recent review on the latter topic can be found in [
33].
DBD of proton-rich isotopes is also possible and is currently in the early stages of experimental exploration [
34]. There are three such DBD modes [
35,
36]: the double-positron emitting (
) mode, the single electron capture with coincident positron emission (
) mode, and the double electron capture (
) mode. Experimental measurements of these modes have received less attention due to their longer half-lives compared to the
decay mode, mostly due to their lower
Q-values. However, their distinct signatures might help in their detection, especially if coincidence trigger logic can be employed [
34]. There are some positive indications of the
mode for
130Ba and
132Ba from geochemical measurements [
37,
38,
39] and for
78Kr [
40,
41]. Recently, the XENON1T collaboration reported the direct observation of the
mode in
124Xe [
42,
43].
From the theoretical point of view, the pioneer estimates of the decay rates for
ECEC have been obtained by Primakoff and Rosen [
44,
45], followed up by the ones made by Vergados [
46] and by Kim and Kubodera [
47]. In these early stages, the predictions were based on a non-relativistic treatment for the captured electrons. Doi and Kotani were the first to present a detailed theoretical formulation for the process and to include the relativistic effects [
35]. Still, the calculations are based on the atomic structure of a point-like nucleus, for which the electron bound states are known analytically. At the same time, Boehm and Vogel presented the results for some selected
ECEC cases, but not many details of the calculations are provided [
48]. Recently, the treatment of the atomic screening was improved using a Thomas–Fermi model for an atomic cloud [
49,
50,
51]. The diffuse nuclear surface effects have also been accounted for through a realistic charge distribution inside the nucleus [
50,
51].
Although ECEC, by its very nature, stands at the interface between nuclear and atomic physics, some aspects regarding the atomic structure calculations have been overlooked or treated simplistically in previous investigations. In this paper, we employ the Dirac–Hartree–Fock–Slater (DHFS) self-consistent framework, providing an enhanced description of atomic screening for the systems involved in the decay and a more rigorous estimation of the binding energies for the captured electrons. We mention that the DHFS atomic potential includes the finite nuclear size and diffuse nuclear surface effects. Furthermore, we extend our analysis to include all s-wave electrons available for capture, surpassing the K and orbitals considered in prior studies.
We demonstrate that, for light atoms undergoing the ECEC process, the decrease in the decay rate resulting from more realistic screening is balanced out by the increase associated with opening the capture from higher orbitals than K and . However, this balance is not observed for medium and heavy atoms, where the latter improvement in our model leads to a considerable increase in the decay rate, up to 10% for the heaviest atoms. For all cases undergoing ECEC, we provide the capture fractions for the first few dominant channels. We specifically address the low Q-value ECEC transitions of 152Gd, 164Er, and 242Cm, for which the capture of both K shell electrons is energetically forbidden. Finally, with the updated phase-space values, we reexamine the effective nuclear matrix elements and compare their spread with those associated with .
3. Results and Discussions
The PSFs for captures from
K and
shells only,
, calculated for the
ECEC cases previously investigated, are shown in
Table 2. Here, we have assumed the same
Q-values as in [
49]. Our results are consistently lower by about 5% than the values reported in [
49], with a couple of exceptions observed in
Mo and
W. Our results generally fall within 70% of the values reported in [
50,
51], with a noticeable dependency on
A: for low mass numbers, our PSFs are lower than those in [
50,
51], but as the mass number increases, the factors in this work approach and even exceed the values reported in that reference, suggesting a discrepancy in the underlying models. In contrast, the
values derived in this work exhibit a consistent 10–20% reduction compared to the ones provided in [
35], which could be attributed to the fact that, in [
35], the screening effect is not accounted for. We chose not to display the
values from [
48], since it does not employ the same definition of the PSF due to a different separation of the decay rate.
In
Table 3, we present the total PSFs
, as well as the capture fractions for selected shell pairs, for all nuclei for which
ECEC is energetically possible, and EC is energetically forbidden. When comparing with one of the most precise calculations available [
49] (column four of
Table 2), one can see the following trend. For light atoms, the more rigorous treatment of atomic screening balances out the effect of accounting for all
shells. The only exception is
Ca, where the PSF value computed using all
shells is higher by about 7.5% than the one in [
49]. The difference is induced by the interplay between the binding energies and
Q-value in the PSF integral (see Equation (
9)). For medium and heavy atoms, there is an increase in the decay rate compared to results from [
49], almost linear in
Z, reaching about 10% for the heaviest atoms.
The low
Q-value transitions of
Gd,
Er and
Cm exhibit an interesting behavior. In these cases, both the
and the
captures are energetically forbidden, and the highest contribution to the total PSF is given by the
,
and
captures. The values of the PSFs and capture fractions for these low
Q-value transitions are given in
Table 4.
The inverse of the PSF is proportional to the half-life for each transition (see Equation (
7)). This quantity is plotted in
Figure 2 for all nuclei shown in
Table 3 versus atomic number. One can see a decrease in the
Q-value as the atomic number increases.
Finally, we investigate the effective matrix elements for the
process of
Kr,
Xe,
Ba and
Ba. These can be obtained as
using the experimental half-lives and the PSFs from
Table 3. The effective NME values are presented in
Table 5. It should be noted that, in the case of
Ba, there is also an experimental half-life measurement [
38]. The authors state that the obtained value is tentative, and indeed, we obtain an effective NME value that is more than one order of magnitude larger than the others. Hence, we employ the half-life limit from [
37]. Future measurements might clarify the situation, but we note that, in general, geochemical measurements tend to underestimate the half-lives, leading to overestimated effective NMEs. This may be a consequence of the difficulty in identifying the relevant production channel of the final atom [
34]. Another interesting hypothesis is the variation in the weak interaction constant with time [
62,
63,
64].
In
Figure 3, we compare the values from
Table 5 with the corresponding
effective nuclear matrix elements. These are defined similarly to the ECEC effective NMEs, but replacing isospin-lowering with isospin-rising operators in the definition of the Gamow–Teller and Fermi matrix elements. We note that effective matrix elements from both processes span similar ranges.
4. Uncertainties and Further Improvements
We first perform a sensitivity analysis on the input parameters, namely the
Q-value, the average energy of the excited
states
and the nuclear radius. The relative variations in the PSFs due to each parameter are summarized in
Table 6, with the
Q-value having the highest influence. However, we note that modern determinations of this quantity are highly accurate and the absolute values of the PSFs are practically insensitive to the choice of
Q-value. The sensitivity of the PSFs with respect to
and the nuclear radius are much smaller, as noted previously in [
49].
Another source of uncertainties is related to the choice of the DHFS model for atomic structure computations. The accuracy of the binding energies obtained in the DHFS approach is around 1% relative to the experimental values [
58]. The binding energies enter in the PSFs definition through the integration limit, through the energies of the emitted neutrinos, and through the
and
factors, always as additive quantities to the
Q-value. Consequently, this uncertainty has a negligible effect as it translates to
relative variation in the
Q-value of the process. Low
Q-value transitions are an exception, and the uncertainty in the binding energies is dominant. The values of the wave functions on the nuclear surface can also introduce some uncertainty in the PSFs. This effect was studied in [
58] by comparing the Coulomb amplitudes (proportional to the wave function values on the nuclear surface) in the DHFS model with the ones obtained through more complex models. The study found that Coulomb amplitudes obtained in the DHFS framework agree within 0.25% with ones obtained through the more refined Dirac–Hartree–Fock (DHF) model for the
and
shells and for atomic numbers above 20. Considering this value as uncertainty within our model, we evaluate the relative error in the PSFs due to the wave functions alone to be 1%, since the wave functions enter the PSF definition to the fourth power. We note that this uncertainty is a systematic one, the Coulomb amplitudes in the DHFS model always overestimating the ones obtained in the DHF model.
Besides the uncertainties discussed above, one may expect variations in PSFs from improving the overall model of the
ECEC process. Firstly, the summed neutrino energy determination has been obtained through a usual approximation (see Equation (
6)), but it can be more precisely determined through Equation (
5). Consequences are most important in low
Q-value transitions. For example, the DHFS framework predicts
keV for
Sm. Consequently, the
process of
Gd is energetically allowed (in contrast to the results shown in
Table 3), as it was also found in [
67,
68,
69] when studying the resonant neutrinoless double electron capture. Secondly, the Pauli blocking of the decay of the innermost nucleon states is not accounted for in our model. This aspect can be improved by averaging the bound electron wave function, weighted with a realistic nuclear charge distribution [
65,
70]. From preliminary results, we expect an increase of a few % in the PSFs due to the Pauli blocking effect.
5. Conclusions
In this paper, we have conducted a systematic study of atoms undergoing the ECEC process. Despite the fact that ECEC, by its very nature, resides at the interface between nuclear and atomic physics, certain aspects related to atomic structure calculations have been either overlooked or treated simplistically in previous investigations. In our model, we introduced two enhancements to the atomic part calculations of double electron capture transitions. Firstly, by employing the DHFS self-consistent framework to describe the initial atomic system, we refined both the distortion of bound wave functions due to atomic clouds and the estimation of binding energies for captured electrons. Secondly, we extended our consideration beyond previous studies, where only K and orbitals were included, by also allowing captures from the outer orbitals of the initial atoms.
With our improved model, we have updated the phase-space values for all atoms undergoing the ECEC process. For light atoms, we observed almost no differences compared to a previous model, where a simplified atomic screening and only captures from K and orbitals were considered. This lack of difference is associated with a cancellation of the decrease in the decay rate from a more precise screening treatment with the increase associated with the possibility of captures occurring from higher orbitals. In contrast, for medium and heavy atoms, we observed an increase in the decay rate, almost linear in Z, reaching about 10% for the heaviest atoms. In the systematic study, we also provided capture fractions for the first few dominant partial channels and separately addressed the low Q-value ECEC transitions of Gd, Er, and Cm, for which the capture is energetically forbidden. Finally, we demonstrated that the effective nuclear matrix elements for both ECEC processes and decays span similar ranges.